Dwarfs and giants: the dynamic interplay of size-dependent cannibalism and competition - Chapter 6 General discussion

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Dwarfs and giants: the dynamic interplay of size-dependent cannibalism and

competition

Claessen, D.

Publication date

2002

Link to publication

Citation for published version (APA):

Claessen, D. (2002). Dwarfs and giants: the dynamic interplay of size-dependent cannibalism

and competition. UvA-IBED.

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Generall discussion

6.11 Introduction

Thiss chapter is essentially the continuation of the General introduction (Chapter 1).. It consists of four parts. First, the two questions raised in section 1.3 are reconsideredd in the light of the results in this thesis. Second, the results and the mechanismss that cause them are discussed in more detail, comparing the results off two different models (the model of Chapters 2 and 3 and the model of Chapter 4).. Third, the comparisons between model predictions and empirical data that are madee in this thesis are reviewed. Finally, the results of this thesis are put in a widerr perspective, addressing the question whether this thesis has implications for ecologicall theory, experiments and/or fisheries management.

6.22 Questions revisited

Inn section 1.3 two questions have been raised: (a) What effect(s) may cannibalism havee on population dynamics? (b) What mechanisms or aspects of cannibalism causee these effects? In the absence of cannibalism our models predict generation cycless caused by size-dependent competition (and a juvenile delay). Introducing cannibalismm can have a number of consequences, which are listed below; the as-pectss of cannibalism (cf. section 1.2.1) that cause each effect are given in italics:

(i)) Stabilisation (i.e., dampening) of competition-driven generation cycles

mortality. mortality.

(ii)) Regulation of population size; occurs in the more or less stable, cannibal-drivenn population state which results from effect (i) mortality. (iii)) Coexistence of a 'dwarf' size class and a 'giant' size class in

competition-drivenn generation cycles; this size-dimorphism corresponds to a bimodal populationn size distribution, and may happen if effect (i) fails (temporarily)

mortality,mortality, gain, size-dependent interactions, competition.

(iv)) The existence of two population states, one 'stunted' and one 'piscivorous'. Inn the stunted state individuals remain small and mainly consume alternative resource.. The piscivorous state is characterised by a very wide population sizee distribution with the largest individuals consuming conspecifics exclu-sivelyy (i.e., the piscivory niche) gain, size-dependent interactions. (v)) For the same conditions, coexistence of a 'stunted' and 'piscivorous'

popu-lationn state, i.e., bistability mortality, gain, size-dependent interactions. Finally,, we may speculate that the ontogenetic niche shift from feeding on an al-ternativee resource to feeding on conspecifics has the potential to induce

(vi)) Evolutionary branching gain, size-dependent interactions, competition. Itt should be kept in mind, however, that the result of evolutionary branching (Chap-terr 5) was obtained for an ontogenetic shift between heterospecific resources. It re-mainss to be shown that this is also possible if the niche shift involves cannibalism. Resultss from interspecific predation do not necessarily carry over to intraspecific predationn (e.g., section 1.2.2).

AA number of the listed effects of cannibalism on population dynamics have not beenn found before. These are the dwarfs-and-giants dynamics, the piscivory niche, andd the 'Hansel and Gretel' effect as a possible explanation of bistability. Interest-ingly,, all three relate to the energy gain of cannibalism, and therefore emphasise thee important distinction between infanticide and cannibalism.

Thee other effects of cannibalism that we have found had already been de-scribedd before. Population regulation by cannibalism has been observed in models byy Ricker (1954), Gurtin and Levine (1982), Diekmann et al. (1986) and Cush-ingg (1992). The stabilising effect of cannibalism was already found by Cushing (1991),, Kohlmeier and Ebenhöh (1995) and van den Bosch and Gabriel (1997). Bothh effects are caused by the cannibalism-induced mortality of victims, which iss the single aspect of cannibalism that is incorporated in all models of the dy-namicss of cannibalistic populations (Table 1.1). Cannibalism-induced bistability wass found by Fisher (1987), van den Bosch et al. (1988) and Cushing (1992). In alll three models bistability results from the presence of a positive effect of canni-balism.. In the model of Fisher (1987) the positive effect of cannibalism is indi-rectt and mediated through competition for shared resources. In his model, a high cannibalisticc mortality rate leads to a high density of alternative food, and hence largee cannibals that cause high cannibalistic mortality. In the other two models the positivee effect of cannibalism is the direct cannibalistic energy gain. Apparently, cannibalismm can give rise to bistability in at least two different ways; via shared resourcess and through cannibalistic gain. In our case bistability is closely asso-ciatedd with the size-dependent nature of the cannibalistic interaction (the Hansel andd Gretel effect, see section 6.3.3). This effect highlights that the population size distributionn itself may modify the balance of the cost and benefits of cannibalism. Itt should be noted that in the list of the effects of cannibalism displayed above thee details of the individual-level model, such as the type of functional response andd energy allocation, do not appear as explanatory mechanisms. Rather, different aspectss of the ecological interactions between individuals are forwarded as such:

thee size-dependent nature of cannibalism and competition, the energy gained by cannibalss and the mortality suffered by victims. In the next section it is investigated too what extend the results depend on the ecological interactions alone, or whether theyy are influenced by the specifics of the bioenergetics.

6.33 Bioenergetics versus interactions

Thee aim of this section is to investigate whether the results obtained in chapters 2,, 3 and 4 depend on the specific assumptions of the individual-level model, in particularr the channelling of energy to growth, maintenance and reproduction, or whetherr they are intrinsic to the ecological interactions between individuals. In the latterr case the results have more general implications, as many different cannibal-isticc species, which differ by the specifics of their bioenergetics, engage in similar size-dependentt cannibalism and size-dependent competition.

Too this end we compare the results obtained with two size-structured popula-tionn models which differ in several important aspects, but which both incorporate dependentt cannibalism as outlined in section 1.4. Also, in both models size-dependentt competition emerges from the size-dependent scaling of vital rates and thee exploitative foraging on a dynamic, alternative resource. A detailed description off the first model is presented in chapters 2 and 3, of the second model in chapter 4.. Here we limit ourselves to the description of the differences and similarities in termss of the underlying, biological assumptions.

6.3.6.3. J Different bioenergetics

Inn the first model mature individuals are assumed to reproduce in synchronised pulsess at the beginning of each growing season. Consequently the population consistss of discrete age cohorts. Within growing seasons the state of individuals changeschanges continuously depending on resource levels and their body size. At re-production,, individuals are assumed to convert their accumulated gonad mass into neww offspring. Hence it is necessary to keep track of the amount of accumulated gonadd tissue, as well as somatic body mass. Moreover, in this model individuals aree assumed to be able to starve away part, but not all, of their body mass when the energyy requirements for body maintenance exceed the assimilation rate. In order too incorporate both accumulated gonads and the possibility of starvation into the model,, the state of individuals is characterised by two state variables; these are referredd to as irreversible body mass and reversible body mass, respectively. Go-nadd tissue is assumed to be a fraction of the reversible body mass. The rules for channellingg of energy in this model are summarised graphically in Fig. 6.1.

Byy contrast, in the second model individuals are assumed to reproduce contin-uously.. As a result, the emerging population size distribution is also continuous. Continuouss reproduction allows for a much simpler individual-level model, since theree is no need to keep track of accumulated gonad tissue, nor (explicit) energy reservess for starvation. Rather, the assimilation rate can be channelled

instanta-T3 3 C C -O O

3 3

,'f ,'f

irreversiblee body mass

Figuree 6.1: Energy channelling and growth in the pulsed model. Arrow (1): In-dividualss are born with a fixed amount of irreversible body mass (x = x;,) and reversiblee mass such that y/x = qj. As they grow, the ratio y/x approaches (or remainss at) the maximum ratio for juveniles, qj. (2): Upon reaching x = Xf, theyy mature. If they are not starving, the ratio y/x now approaches the maxi-mumm ratio for adults, qA- (3): At the start of a growing season, adults convert all theirr reversible body mass in excess of y = qjx into offspring. Resource densi-tiess permitting, they start growing again after reproduction. (4): If, however, the assimilationn rate is insufficient to cover metabolism, they starve away reversible bodyy mass. Below the ratio y/x — q$ their mortality rate increases steeply with decreasingg y/x.

neouslyy to reproduction, metabolism and somatic growth. Starvation can then be incorporatedd by decreased growth and reproduction rates, and possibly a negative somaticc growth rate. In the model in Chapter 4, we assumed that a fixed fraction (11 — K) of the energy assimilation rate is allocated to reproduction, whereas the otherr fraction (AC) is used for metabolism and somatic growth. This type of energy channellingg has been referred to as the «-rule (Kooijman and Metz, 1984; Kooij-man,, 1986, 1993, 2000, 2001), and falls in the category of assimilation allocation, ass opposed to net-production allocation (Gurney and Nisbet, 1998). In the continu-ouss model the state of individuals is hence characterised by a single state variable; length. .

Ass described above, the two models differ most fundamentally in their energy channelling.. Furthermore, the two models incorporate different functions to de-scribee the relation between body size and vital rates, such as handling time and metabolicc rate. The pulsed model was designed to be parameterised with

experi-mentall data on individual-level performance. In the continuous model most of the parameterisedd functions of the pulsed model have been approximated by simpler functions.. For example, in the pulsed and continuous models the metabolic rate is givenn by, respectively:

M{x,y)M{x,y) = p1 (x + y)P2

and d

M(l)M(l) = pl3

wheree x and y are irreversible and reversible body mass, respectively (Fig. 6.1), p\ andd p2 are two allometric constants, / is body length and p an allometric constant.

Bothh models assume a Holling type II functional response.

6.3.26.3.2 Similar interactions

Thee ecological interactions in the two models are essentially the same. Size-dependentt competition for the alternative resource results from the size-scaling off foraging rate and metabolic requirements. It is best captured by the relation betweenn the minimum resource density required for growth (the 'critical resource density')) and body size. In both models, this is an increasing function of body size, implyingg that smaller individuals are competitively superior to larger ones (Pers-sonn et al., 1998). Both models assume semi-chemostat dynamics of the alternative resource. .

Size-dependentt cannibalism is modelled in exactly the same way in both mod-els.. The cannibalistic attack rate of a cannibal of length c and a victim of length v iss given by:

AAcc{c,v)=C{c,v)=Cmaxmax(c)T(c,v)(c)T(c,v) (6.1)

wheree the function Cmax(c) describes the maximum possible attack rate of a

can-niball of length c, attained for optimal victim sizes only, which is given by:

CmaCmaXX{c)=Pc{c)=Pcaa ( 6 . 2 )

Thee function T(c,v) takes into account the effect of suboptimal victim sizes and iss given by:

\wWc\wWc XSc<v<<j>c

TT

^^vv)=\Tr)=\Tr==WcWc ti<!>c<v<ec (6.3)

[[ 0 otherwise

whichh is a tent-shaped function that takes values between one when the ratio be-tweenn victim and cannibal length is optimal (v/c = 0), and zero at the lower and upperr limits of the cannibalism window (v/c — 5 or e, respectively). Outside this windoww T(c, v) is zero.

Withh respect to how cannibalism is incorporated, the only difference between thee two models is in the parameter value of a, which is assumed to be 0.6 in the pulsedd model and 2 in the continuous model. Note that for individual of 140 mm,

CmaxCmax in the continuous model and in the pulsed model differ by a factor 1000.

Forr example, Cmax in the continuous model with (3 = 0.8 is comparable to Cmax

inn the pulsed model with ft — 800. A second difference results indirectly from thee different timing of reproduction. Whereas in the continuous model (at least some)) victims are available continuously, in the model with pulsed reproduction thee availability of victims varies strongly within seasons.

6.3.36.3.3 Similar results ?

Thee list of effects of cannibalism (section 6.2) does not differentiate between the twoo different models with which the results were obtained. Below we discuss the firstt five effects on that list, and compare the results obtained with the pulsed and thee continuous models. The comparison will show to what extend the results are determinedd by the ecological interactions or by the bioenergetics.

(i)) Stabilisation

Onee of the most striking results obtained with the pulsed model is the critical de-pendencee of population dynamics on the lower limit of the cannibalism window, Ö (Chapterr 3). Above a critical value of 6 competition-induced, single-cohort cycles aree found, whereas below it either dwarfs-and-giants dynamics are found or can-nibalismm stabilises dynamics in which case a more or less stable population size distributionn is found. A remarkably similar result is found with the continuous model,, as is shown in Fig. 6.2. Cannibal-driven dynamics are found for small Ö, dwarfs-and-giantss dynamics for intermediate 5, and the equivalent of single-cohort cycless for high 5. There are, however, interesting differences. First, dwarfs-and-giantss dynamics are found for much higher values of S (up to 6 ~ 0.15). Second, thee critical value of S below which cannibal-driven dynamics occur is also higher inn the continuous model (Ö w 0.095). Third, in between these two critical values off 6, both dwarfs-and-giant dynamics and single-cohort dynamics can be found, dependingg on initial conditions. Bistability of these types of population dynamics iss not found with pulsed reproduction.

Thee explanation of these differences lies in the timing of reproduction and the consequencee it has on the resulting population size distribution. Clearly, pulsed reproductionn results in a pulsed population size distribution where each pulse is a discretee age class. Continuous reproduction gives rise to a continuous size distri-bution.. It should be noted, however, that even with continuous reproduction the size-distributionn may consist of discrete size classes, separated from each other by 'empty'' size intervals (i.e., no individuals in that size interval). The competition-inducedd population cycles that result with a high 6 (e.g., S > 0.15 in Fig. 6.2) aree the equivalent of 'single-cohort cycles' discussed in Chapter 3. Competition betweenn newborns and larger individuals is so strong that, regularly, a high density off newborns causes starvation mortality of large juveniles and adults. As a con-sequence,, the population consists of discrete age classes or 'generations'. In such populationn cycles periods without any reproduction due to the absence of adults

(a)) (b) ~£~£ 800 ^^ 600 en n c c _QQ 400 E E ^^ 200 X X

JJ 0

Figuree 6.2: The effect of the lower limit of the cannibalism window, S, on popula-tionn dynamics, characterised by the maximum length at time of censussing. Both bifurcationn diagrams are constructed from simulations (i.e., not continuation) with:

(3(3 = 500, e = 0.5, <f> = 0.2. In both (a) and (b), from low to high 5 the type of

populationn dynamics are cannibal-driven, dwarfs-and-giants (obvious from gigan-ticc lengths), and single-cohort cycles, respectively, (a) Pulsed model (S\ = 0.077); censussedd at the first day of each growing season. Single-cohort cycles are evi-dentt from the regular 7-year cycles, (b) Continuous model (S\ — 0.11); output generatedd at local minima and maxima of the alternative resource. The arrow indi-catess the lowest value of 5 (= 0.095) for which single-cohort cycles are found. In thee interval S e (0.095,0.15) both dwarfs-and-giants dynamics and single-cohort cycless are possible, depending on initial conditions.

aree alternated by periods of continuous reproduction. Because there is a time in-tervall between maturation of the first individuals in a generation and the death of thee last ones, the next generation will consist of a range of ages and sizes. Smaller individualss grow faster, however, and therefore the width of the size distribution off a generation decreases over time. Therefore, continuous reproduction may lead too either a continuous or a nearly pulsed size distribution, which may explain the off bistability of single-cohort cycles and dwarfs-and-giants dynamics.

Inn Chapter 3 it is shown that in the pulsed model the expected value of S below whichh cannibalism can stabilise single-cohort cycles is

wheree L\ is the length at first reproduction and Ls the length of newborns at the

momentt the last adults die of starvation (see section 3.3.1). In the continuous modell the maturation size is assumed to be 115 mm. From time series of single-cohortt cycles in the continuous model we can determine that the largest adults becomee only 116.5 mm.. Furthermore, we find that the starvation period is rs w 16

thee lengths of the offspring generation are distributed between 7 mm (the length att birth) and 12.5 mm. If we substitute Ls = 12.5 mm and L\ = 116.5, the

rulee derived for the pulsed model hence predicts a critical value of Si = 0.107 for thee continuous model. Note, however, that the recruits have a length distribution ratherr than a single length, and therefore this estimate of S\ should be considered ass an upper estimate of the critical value. As an alternative, we may substitute the

averageaverage length of the recruits (10.5 mm) for Ls, which gives Si = 0.09. If we

comparee these two estimates of Si with Fig. 6.2, it appears that they are indeed veryy close to the lower boundary of single cohort dynamics (S = 0.095, in Fig. 6.2bb indicated with an arrow). Although we cannot make a precise estimate of

5i5i in the continuous model, the estimates (i.e., 0.107 and 0.09 from the rule and

0.0955 from the simulation) are consistently higher than in the pulsed model, where

SSxx — 0.077. This difference may be a result of differences at the bioenergetic level;

adultss starve to death earlier in the continuous model, but newborns grow faster. Whilee this may explain the quantitative difference between the critical values of

SS in the pulsed and continuous models, the similar, drastic change of population

dynamicss around the critical S is determined by the ecological interactions. Surprisingly,, with continuous reproduction dwarfs-and-giants dynamics are foundd for 8 values above the critical value, rather than below it as is the case in the pulsedd model (Fig. 6.2a). Why this is the case is discussed under point (iii).

(ii)) Population regulation

Inn the absence of cannibalism the size-structured population does not grow out of

boundss because survival, growth and reproduction of individuals are limited by the amountt of alternative food they consume. Competition for the alternative resource hencee regulates the population.

Inn the cannibal-driven population state which results with sufficiently low 5 as discussedd under point (i), the cannibalistic mortality rate of small individuals is veryy high (e.g., Fig. 2.4). Despite a high population birth rate the severe mortality resultss in a low number of (competitively superior) small individuals. Competi-tionn for the alternative resource is hence weak and juveniles grow quickly due to thee high level of alternative food. Another effect of weak competition is the lack off starvation mortality, resulting in the coexisting of old and young individuals. Althoughh adult reproduction is determined by both the amount of alternative food andd the amount of conspecific food, cannibalism indeed acts as a mechanism of populationn regulation in these circumstances because it is the main cause of mor-tality.. With non-fixed point dynamics, occasional high pulses of newborns are also killed-offf and regulated by the existing size class of cannibals.

Thee main aspects of the cannibal-driven population state, i.e., fast juvenile growth,, high juvenile mortality and coexistence of many age classes, are observed inn cannibal-driven dynamics in both the pulsed model (Fig. 2.4, Fig. 3.5) and the continuouss model (Fig. 4.2, Fig. 6.2).

(a) ) (b) ) E E £ £ c c <D <D C C O O r: : _{o o} O O o o O O 03 3 0) ) 600 600 00 450 900 1350 1800 0 450 900 1350 1800 600 0 400 0 200 0 0 0 101 1 10° ° 10"' ' IQ"2 2

Timee (days)

Figuree 6.3: Time series of dwarfs-and-giants dynamics, characterised by growth trajectoriess (upper panel) and resource density (lower panel), (a) Pulsed model withh parameters: (3 = 400, S = 0.056, e = 0.42, <f> = 0.15. (b) Continuous modell with parameters (3 = 300, S - 0.1, e = 0.45, <f> = 0.2. Note that for the pulsedd model growth trajectories of all present cohorts are depicted, whereas for thee continuous model only a number of representative samples is shown.

(iii)) Dwarfs-and-giants dynamics

Anotherr striking result from the pulsed model are the dwarfs-and-giants cycles (Chapterr 2, 3, Fig. 6.3a), found in a range of Ö values in between the critical val-uess öi and 52 (Chapter 3, Fig. 6.2). This type of dynamics is characterised by

competition-inducedd population cycles, in which the (numerically) dominant co-hortt cannibalises the first pulse of offspring whereas the next pulse of offspring outcompetess the adult size class. During their first year, the few survivors of the firstfirst pulse of offspring (referred to as giants) grow fast until the resource is de-pressedd by the next cohort. Their body size allows them to cannibalise the later offspringg (referred to as dwarfs). The dwarfs grow slowly due to intense compe-titionn for the alternative resource. They serve as cannibalistic food to the slightly olderr but much larger giants, which continue to feed on the dwarfs (Fig. 6.2a).

Fig.. 6.3b shows that the continuous models exhibits very similar dynamics. Thiss may come as a surprise, since the above described mechanism is hard to imaginee with continuous reproduction. If, with continuous reproduction, small adultss are able to cannibalise newborns, this is sufficient to prevent starvation

mor-talityy of adults, and thus sufficient to lead to cannibal-driven dynamics. This is illustratedd by the occurrence of cannibal-driven dynamics up to the critical value

SiSi (Fig. 6.2b). One might expect, therefore, that dwarfs-and-giants dynamics are

impossiblee in the continuous model. Indeed, starting from single-cohort cycles withh 6 = 0.19, one can gradually lower 5 without changing population dynamics downn to 0.095 (the arrow in Fig. 6.2b), at which value the dynamics change to cannibal-drivenn dynamics, without ever spotting a giant cannibal. However, start-ingg from the cannibal driven dynamics with 6 = 0.05, and gradually increasing

6,6, it is impossible to miss the dwarfs and giants dynamics. It appears that in the

single-cohortt cycles (in the range <5 « 0 . 1 . . . 0.15) there is a 'niche' for giant can-nibals,, but that the individuals of the dwarf cohorts and their offspring cannot enter it.. As discussed under point (i), the size distribution of the population in single-cohortt cycles is nearly pulsed, and therefore reproduction occurs discontinuously inn time. While an individual can enter the 'giant cannibal niche' if it is born at the rightt moment (that is, in a certain phase of the cycle), this does not occur due to the absencee of adults in this phase of the cycle. If, however, by a perturbation some individualss of adequate size are added to the system, they thrive well because they cannibalisee the abundant dwarf cohort. Once mature, the individuals in the giant canniball niche reproduce during a prolonged period. Their reproductive period includess the timing required for entering the next giant cannibal niche, and there-foree the giants can form a persisting subpopulation. Interestingly, the giant cohorts inn the continuous model are not the offspring of dwarf cohorts, which contrasts withh the pulsed model, in which giants and dwarfs are nearly 100% full-sibs from dwarf-sizedd parents. Rather, every generation of giants is produced by the previ-ouss generation of giants (while most dwarfs are produced by dwarfs). It should be noted,, however, that 'genetically' segregated subpopulations of dwarfs and giants occurr only in very stable dwarfs-and-giants cycles, such as depicted in Fig. 6.3b. Forr many parameter values the dynamics are not regular, and mixing occurs fre-quently.. Unstable dwarfs-and-giants dynamics are expected if the subpopulation inn the giant cannibal niche becomes abundant enough to have a dynamic impact onn the dwarf cohort.

Inn summary, the occurrence of dwarfs-and-giants dynamics depends critically onn the dynamic interplay of size-dependent cannibalism and competition (i.e., in-teractions,, not bioenergetics). Time series of this type of dynamics, obtained with thee two models, are very similar. Yet, there are subtle but important differences, whichh can be attributed to the mode of reproduction (pulsed vs. continuous) and thee resulting population size distribution.

Alreadyy in the discussion of Chapter 2 a continuous-time model has been pre-sentedd (Fig. 2.9), which is similar to the continuous model in Chapter 4 except that itt assumes a linear functional response. The occurrence of dwarfs and giants hence seemss independent of the shape of the functional response as well.

(iv)) Stunted and piscivorous population states

AA conspicuous result found with the pulsed model is the dependence of the popula-tionn size distribution on the upper limit of the cannibalism window, e (Chapter 3). Inn bifurcation runs the maximum size attained in the population increases quickly fromm around 18 cm to over 30 cm around a critical value of e (Fig. 6.4a). A sim-ilarr result is found with the continuous model (Fig. 6.4b), although the effect is moree drastic because the equilibrium curve is folded. In both models we can dis-tinguishh between a 'stunted' population state and a 'piscivorous' population state. Thee stunted state is characterised by small ultimate size, little gain from canni-balism,, and a high alternative resource density. Inversely, the piscivorous state iss characterised by large ultimate sizes, a high gain and a low resource (although nott as low as in the single-cohort cycles or dwarfs-and-giants cycles). The pis-civorouss state is further characterised by the presence of individuals that consume conspecificss only, because they are too large to consume the alternative resource (i.e.,, individuals in the piscivory niche).

Inn order to characterise the role of cannibalism in the different equilibrium statess quantitatively, Fig. 6.4 depicts the fraction of population fecundity that is derivedd from cannibalistic food intake (which equals the fraction of a single in-dividual'ss expected life time reproduction that is derived from cannibalistic food intake),, the probability of an individual to escape cannibalism, and the probability off an adult to fall victim to cannibalism. Of these three aspects, the first relates too the benefit of cannibalism (via the direct energy gain), whereas the latter two relatee to the cost of cannibalism (via additional mortality). These data allow us too address the following question: is the stunted (resp., piscivorous) population statee in the pulsed model similar to the stunted (resp., piscivorous) state in the con-tinuouss model? In other words, are the stunted and piscivorous population states comparablee in the two models? As pointed out above, with respect to the ulti-matee size the stunted and piscivorous states are similar in the two models. Fig. 6.44 shows further that the stunted state in both models is characterised by a small (butt increasing with e) contribution of cannibalistic energy intake to reproduction, aa high probability to escape cannibalism, and a negligible probability for adults to falll victim to cannibalism.

Thee figure shows that with respect to the piscivorous state, the pulsed model deviatess in two ways from the continuous model. Firstly, in the pulsed model thee alternative resource remains an important contribution to population fecun-dityy even in the piscivorous state, whereas in the continuous model reproduction derivess for nearly 100% from cannibalistic food intake in the piscivorous state. Apparently,, small adults that still have a significantly planktivorous diet deliver thee main contribution to population fecundity in the pulsed model. This differ-encee corresponds to a difference in population size distribution, which is roughly exponentiallyy distributed in the pulsed model (Fig. 3.8) and U-shaped in the con-tinuouss model (Fig. 4.4). In the latter situation large adults dominate reproduction, whereass in the former one the small adults dominate reproduction. Secondly, the probabilityy to escape cannibalism does not differ spectacularly between the two

Figuree 6.4: The effect of the upper limit of the cannibalism window, e. (a) Pulsed model,, based on simulations (cf. Fig. 3.7), cencussed on first day of the growing season,, with 0 = 200, 6 = 0, 4> = 0.2. (b) Continuous model, based on continu-ationn with /3 — 0.8, 6 = 0.03, 4> = 0-2. Top panel: maximum length in the popu-lationn (at the time of censussing). Second panel: fraction of population fecundity thatt is derived from cannibalistic food intake (which equals the fraction of a sin-glee individual's expected life time reproduction that is derived from cannibalistic foodd intake). Third panel: the probability of an individual to escape cannibalism.

BottomBottom panel: the probability of an adult to fall victim to cannibalism. Note that

inn the continuous model the values on the curve of stunted equilibria and on the curvee in between the two fold bifurcations lie on the axis.

populationn states in the pulsed model.

Overall,, Fig. 6.4 suggests that the results from the two models are qualitatively similar.. The stunted state is associated with (relatively) low benefit and low cost off cannibalism, whereas in the piscivorous state both the benefit and the cost of cannibalismm are high. (Note that the net benefit of cannibalism is always negative inn both stunted and piscivorous states, in the continuous model; Fig. 4.5). Yet the quantitativee differences concern two factors which are likely to affect population dynamics:: reproduction and the probability to escape cannibalism (the middle two panelss in Fig. 6.4). In Chapter 3 we argue that the effect of e on population dy-namicss is small because it mainly affects old cohorts with little dynamic influence.

Thee reason why young, cannibalistic cohorts are more important for popula-tionn dynamics in the pulsed model than in the continuous model may be due to the pulsedd nature of reproduction itself. As pointed out above, pulsed reproduction resultss in a pulsed population size distribution (e.g., Fig. 2.5, Fig. 3.8). This has threee important consequences for the cannibalistic interactions. (1) The availabil-ityy of victims varies strongly within a growing season as victims move through the cannibalismm window. For small cannibals, this means that part of the year there is noo conspecific food available and they have to rely on alternative food or, if alter-nativee food is scarce, they starve away reserves. (2) The pulsed availability of food meanss that cannibals are more handling time limited, and hence kill fewer victims perr year than cannibals in a continuously reproducing population. (3) Individuals cannott start cannibalising before reaching the age of one year, when the first cohort off potential victims is produced. With pulsed reproduction the size of one-year-old individualss is between 60-100 mm, in both the stunted and piscivorous population statess (Fig. 2.5, Fig. 3.8). Consequently, individuals are obligatory planktivorous upp to that size. By contrast, in the continuous model individuals start cannibalis-ingg upon reaching length x^/e; with e = 0.5 and xt — 7 mm, this equals 14 mm. Thesee aspects may explain why the adult subpopulation is dominated by small adultss in the pulsed model, rather than large adults as in the continuous model. In turn,, the smaller contribution of cannibalism to the energy balance of small adults mayy explain the quantitative difference observed in Fig. 6.4.

(v)) Bistability of stunted and piscivorous states

Thee most striking difference between the results of the two models is the presence off bistability in the continuous model and its absence in the pulsed model (Fig. 6.4).. In certain parameter ranges, the continuous model predicts that the popu-lationn can either converge to a stunted state or a piscivorous state, depending on initiall conditions. In Chapter 4 we argue that the driving force of the bistability is thee Hansel and Gretel effect. In the continuous model, the net benefit of cannibal-ismm is much higher in the piscivorous state than in the stunted state (Fig. 4.5, and seee Fig. 6.4: large contribution of cannibalism to reproduction). The net benefit of cannibalismm depends on the size distribution of cannibals. Due to the cannibalism window,, if cannibals are larger they have a larger gain from cannibalism because onn average the cannibals consume victims when the victims are larger and contain

hencee more energy. This effect is called the 'Hansel and Gretel' effect (Chapter 4), namedd after the tale which, to our best knowledge, is the first account of the idea too postpone cannibalism until the victim has become more nutritious (Grimm and Grimm,, 1884).

Whyy is this bistability not observed in the pulsed model? With respect to the cannibalisticc interaction, there are two main differences between the two mod-els.. The first is simply the difference in the exponent of the size-scaling of the maximumm cannibalistic attack rate (a = 0.6 in the pulsed model, a — 2 in the continuouss model). Simulation of the pulsed model with a = 2, however, show thatt the pattern depicted in Fig. 6.4a does not change significantly, and hence the valuee of a cannot explain the absence of bistability. The second difference is the pulsedd vs. continuous size distribution, with the three consequences for cannibal-ismm discussed under point (iv). As pointed out above, we argue in Chapter 4 that thee observed bistability in the continuous model may be the consequence of a pos-itivee feedback loop created by the Hansel and Gretel effect. Compared to a small cannibal,, a large cannibal gains more energy from its victims because its victims are,, on average, larger. More energy may result in individual growth. As the large canniball becomes even larger it gains even more energy, as its victims are even larger,, etc. We conjecture that bistability is absent in the pulsed model because withh a pulsed size distribution a cannibal of a larger size does not necessarily have accesss to larger victims, simply because the set of available victim sizes is limited. Thiss conjecture, however, remains to be established more firmly.

Inn summary, we conclude that most results that we have found are independent off the specifics of the individual-level model. In particular, the results that relate too cannibalistic mortality rather than cannibalistic gain are very robust. Where we foundd differences between the continuous and pulsed models, this appears to be duee to the timing of reproduction and the resulting population size distribution, ratherr than due to the allocation of energy or other bioenergetic aspects.

6.44 Testable hypotheses - the link with empirical data

Onee of the aims of this thesis has been to develop testable hypotheses which can bee compared with empirical data. Physiologically structured population models aree defined primarily at the individual level, specifying how the state of an indi-viduall changes with time in response to environmental conditions. This approach allowss one to develop a model which is easily parameterised because model 'ingre-dients'' can be formulated in terms of experimentally measurable quantities, such ass size-dependent attack rates, handling times, and fecundity. This contrasts with populationn models that are formulated at the population level, which are the more commonlyy used models in ecology. These are necessarily more phenomenological andd therefore less testable.

Thee model developed in Chapter 2 is based on Eurasian perch (Percafluviatilis) feedingg on zooplankton. The models in the other chapters are all based on this

modell and its parameterisation, and their results should therefore be comparable withh empirical data on perch population dynamics. However, by changing model parameters,, the model represents the biology of other, related species, such as yel-loww perch (P. flavescens), pikeperch {Stizostedion lucioperca), pike (Esox lucius), roachh {Rutilus rutilus) or arctic char (Salvelinus alpinus). At least qualitatively, ourr model should be able to predict differences between population dynamics of thesee species by comparing results with different parameter values.

Off the chapters in this thesis, Chapter 2 contains the most extensive comparison betweenn model predictions and empirical data. In a comparison of cannibal-driven andd dwarfs-and-giants dynamics with an empirical time series of perch in Lake Abborrtjarnn 3 it was shown that the (population dynamic) mechanism of induction off giant growth is remarkably similar in model and data. Thus, our model offers a populationn dynamic explanation for the occurrence of giant cannibals. Yet, there weree also discrepancies between model results and data, particularly with respect too the dynamics following the induction of giant growth. To understand these discrepanciess is one of the goals of ongoing research (Persson et al, in prep).

Basedd on the analyses in Chapter 3 we can make predictions about differences inn population dynamics in related cannibalistic species. Chapter 3 focuses on the relationn between the size-dependent nature of cannibalism and the emerging pop-ulationn dynamics. A comparison is made between two closely related species, Eurasiann perch and yellow perch. The latter species has a higher value of S and cannibalismm is hence predicted to have a smaller stabilising effect than in Eurasian perch.. Comparison of time series appears to confirm this prediction. Further, the chapterr contains a more speculative discussion of arctic char population dynamics. Thee population model studied in Chapter 5 is simplified to such an extent that itt cannot be readily compared with empirical data. In order to obtain tractable resultss it has been stripped of some biological assumptions such as a juvenile de-layy (but see appendix 5.B). Yet, it contains mechanistic ingredients, such as the size-scalingg of the functional response, which can be parameterised with experi-mentall data. This means that the model predictions can be linked to these param-eters.. However, comparisons with observations should be made with reservation. Testablee predictions cannot be made before a more system-specific model has been developed,, tailored for a particular species of interest.

Essentially,, Chapter 5 shows that the presence of an ontogenetic niche shift inn the life history can result in evolutionary branching. It remains to be shown, however,, that this is also the case if the second resource consists of conspecifics. Inn this case the situation is more complex due to a number of feedback mechanisms betweenn the evolutionary trait and population size distribution. The most obvious differencee with the model in Chapter 5 is that with cannibalism the mortality rate inn the evolving population depends strongly on the evolutionary trait, which almost certainlyy has important consequences.

6.55 This thesis in perspective

Thee 'discovery' of giants (Chapter 2) leads to a search image, i.e., the conspicuous patternpattern of growth curves such as shown in Figures 2.8, 2.9 and 2.10, which allows forr testing of predictions. If another population were found exhibiting a similar growthh pattern (e.g., LeCren, 1992), our model predicts that a very specific se-quencee of population dynamic 'events' would be associated with the acceleration off giant growth. If time series of the population dynamics are available it can be checkedd whether the predicted series of events has occurred or not. Alternatively, iff a lake contains a 'stunted' population, our model predicts that removing the bulk off the cannibal size class results in the emergence of dwarfs and giants. Our model alsoo predicts, however, that the existence of giants is transient, which may limit thee interest of fisheries managers. Thus, our model of size-dependent cannibalism andd competition provides testable predictions. Besides Eurasian perch candidate speciess include yellow perch, arctic char, brown trout and pikeperch. Interestingly, giantt growth patterns have already been described for arctic char (Hammar, 1998) andd ferox trout (Campbell, 1979).

Despitee its potentially transient effect, removing the bulk of the cannibal size classs may be a manipulation by which fisheries managers can transform a commer-ciallyy uninteresting, stunted fish population into one with at least some gigantic individuals,, interesting for sports fishing. Removing the bulk of the cannibals re-ducess the cannibalistic mortality rate of small juveniles. The resulting high density off small juveniles leads to a depletion of the alternative resource, but provides the feww surviving cannibals with sufficient food to become giants. This manipulation standss in sharp contrast to the standard solution for management of stunted fish populations,, which consists of removing the bulk of the small juveniles, in order too release the alternative food, which may increase the growth rate of all remaining individuals. .

Chapterr 3 presents a frame of reference for a comparison of population dynam-icss between different species. We argue that most of the results we find depend on thee cannibalistic and competitive interactions. We show that these interactions de-pendd heavily on the size-dependent nature of cannibalism (i.e., on S and e). Thus, wee can compare the size-dependence of different species and use the results of Chapterr 3 to make predictions about their population dynamics. We have done thiss for two closely related perch species (section 3.4.1). For arctic char the values off 5 and e can be estimated from literature data (e.g., S — 0.15 and e = 0.47 based onn Amundsen, 1994) and together with estimates of Li « 100 — 149 mm and

L(,L(, « 20 mm this leads to the prediction of cannibal-driven (CD) population

dy-namicss (cf. Fig. 3.5). For such between-species comparisons the same candidate speciess as mentioned above are relevant, and relevant size-dependent data from thesee species would prove most useful.

Thee development of the continuation methodology of Kirkilionis et al. (2001) andd the possibility to include infinite-dimensional interaction environments (Chap-terr 4) may improve the level of analysis of physiologically structured population modelss considerably. The development of such techniques for ODEs and discrete

mapss (e.g., Kuznetsov, 1995) has improved the understanding of these classes of models.. From a theoretical point of view, using this method gives more insight inn the underlying structure of population dynamics than numerical simulations. PSPMss tend to show rather complicated bifurcation patterns, which are hard to disentanglee with simulations alone. From an empirical point of view, this develop-mentt allows to scan parameter space of mechanistic population models for testable predictions,, such as bistability. The use of bifurcation diagrams in a manipulative experimentall setting has been shown to be fruitful (Costantino et al., 1997). Fi-nally,, the continuation method may facilitate the study of evolutionary dynamics off size-structured populations (appendix 5.B).

Inn Chapter 4 we speculate that the presence or absence of submerged vegeta-tionn may effectively lead to a high and low value of the lower limit of the predation windoww (S). The mechanism behind this effect would be that vegetation provides aa refuge from cannibalism for YOY fish, effectively increasing the minimum size att which victims can be cannibalised. Decreasing the value of 5 may lead to a catastrophicc bifurcation which is associated with a dramatic reduction in the max-imumm size of fish in the population. Thus, the effect of removing vegetation from aa lake with a population with a wide size distribution may have detrimental ef-fectss for the existence of commercially interesting, large piscivores. Alternatively, thiss analogy suggests that by introducing vegetation, it may be possible to move aa stunted, commercially (and touristically) uninteresting fish population into the piscivorouss state. Although the presence of alternative stable states creates inter-estingg opportunities for fisheries managers (Scheffer, 1998; Scheffer et al., 2001), att the same time they pose a potential problem, as the wrong manipulation may bringg the population in the unwanted, stunted population state.

Chapterr 5 is one of first examples of a mechanistic and hence testable model studiedd within the adaptive dynamics framework so far. The use of physiolog-icallyy structured population models in adaptive dynamics may prove to be very fruitful,, as PSPMs are easily linked to empirical data, and fit very well into the individual-basedd setting of adaptive dynamics. Obviously, our model needs more developmentt before rigorous testing is possible, but nevertheless models like this createe opportunities for testing the different concepts developed in the theoretical frameworkk of adaptive dynamics, which may increase its credibility as a scientific theory. .

Thee comparison of cannibalism in the pulsed model and in the continuous modell (section 6.3) may be a source of testable hypotheses as well. In that section itt is argued that the main differences between the results of the two models stems fromm the different timing of reproduction and the resulting different population sizee distributions. For example, dwarfs-and-gians dynamics are expected below andd above the critical value of Si with pulsed and continuous reproduction, respec-tively.. Although the stunted and piscivorous population states exist with both types off reproduction, bistability is expected only with continuous reproduction. Thus onee could compare of cannibalistic populations with contrasting timing of repro-duction.. Ideally, one would need data on population dynamics of an organism that livess in a gradient of environments from, for example, seasonal to tropical, which

leadss to a gradient from pulsed to continuous reproduction. Potential candidates mayy be found among squid species, some of which are known to be cannibalistic (Sauerr et al., 1992; Santos and Haimovici, 1997). Unfortunately, the commonness off cannibalism among cephalopod species is unknown. Some squid species occur inn a range from tropical to temperate seas, such as Loliolus noctiluca, which occurs fromm New Guinea to Tasmania (Jackson and Moltschaniwskyj, 2001). Although thee life cycle of this particular species may be too short (4 months in tropical pop-ulations;; Jackson and Moltschaniwskyj, 2001) for the environmental seasonality too imose pulsed reproduction, in the class of Cephalopoda suitable species may be foundd to test the different predictions related to the timing of reproduction.

Finally,, in the theoretical context regarding the dynamics of cannibalistic pop-ulations,, this thesis points to a conceptual aspect of cannibalism which has rarely beenn made explicit so far: the distinction between cannibalism and infanticide (or:(or: cannibalism without energy gain). Our results suggest that by neglecting the energyy gain a modeller may miss out some of the most exciting effects of can-nibalism.. Interestingly, 'infanticide' may be a dynamic effect in the sense that dependingg on the state of the population, cannibalism may or may not yield a sig-nificantt gain. This may mislead an ecologist into thinking that the energy gain can bee ignored. Obviously, if this is done the alternative population state, in which the gainn is substantial, can never be predicted.